Thermal Design

1. Thermal Design Heat Transfer Temperature Measurement The prevalence of the number 5.7

2. UCSD: Physics 121; 2011 Why Care about Thermal? • Scientific equipment often needs temperature control • especially in precision measurement • Want to calculate thermal energy requirements • how much energy to change temperature? • how much power to maintain temperature? • Want to calculate thermal time constants • how long will it take to change the temperature? • Want to understand relative importance of radiation, convection, conduction • which dominates? • how much can we limit/exaggerate a particular process?

3. UCSD: Physics 121; 2011 Chief Thermal Properties • Thermal Conductivity •  measured in W/m/K • heat flow (in W) is P = ·T·A/t • note that heat flow increases with increasing T, increasing surface area, and decreasing thickness (very intuitive) • Specific Heat Capacity • cp measured in J/kg/K • energy locked up in heat is: E = cp·T·m • energy stored proportional to T, and mass (intuitive) • Emisivity,  • power radiated is P = AT4

4. UCSD: Physics 121; 2011 Thermal Conductivity of Materials • (copied from materials lecture)

5. UCSD: Physics 121; 2011 Conduction: Heated Box • A 1 m  1 m  2.5 m ice-fishing hut stands in the 10 C cold with 2.5 cm walls of wood • A = 12 m2 • t = 0.025 m •   1 W/m/K • To keep this hut at 20 C would require P = ·T·A/t = (1.0)(30)(12)/(0.025) = 14,400 W • Outrageous! • Replace wood with insulation:  = 0.02; t = 0.025 P = ·T·A/t = (0.02)(30)(12)/(0.025) = 288 W • This, we can do for less than $40 at Target • First example unfair • air won’t carry heat away this fast: more on this later

6. UCSD: Physics 121; 2011 A Cold Finger • Imagine a plug of aluminum connecting the inside to the outside • how much will it change the story? • cylindrical shape, length t, radius R •  = 205 W/m/K • just based on conduction alone, since difference in thermal conductivity is a factor of 10,000, the cold finger is as important as the whole box if it’s area is as big as 1/10,000 the area of the box. • this corresponds to a radius of 2 mm !!! • So a cold finger can “short-circuit” the deliberate attempts at insulation • provided that heat can couple to it effectively enough: this will often limit the damage

7. UCSD: Physics 121; 2011 R-value of insulation • In a hardware store, you’ll find insulation tagged with an “R-value” • thermal resistance R-value is t/ • R-value is usually seen in imperial units: ft2·F·hr/Btu • Conversion factor is 5.67: • R-value of 0.025-thick insulation of  = 0.02 W/m/K is: R = 5.67t/ = 5.670.025/0.02 = 7.1 • Can insert Home-Depot R=5 insulation into formula: P = 5.67A·T/R • so for our hut with R = 5: P  5.67(12)(30)/5 = 408 W • note our earlier insulation example had R = 7.1 instead of 5, in which case P = 288 W (check for yourself!)

8. UCSD: Physics 121; 2011 Wikipedia on R-values: • Note that these examples use the non-SI definition and are per inch. Vacuum insulated panel has the highest R-value of (approximately 45 in English units) for flat, Aerogel has the next highest R-value 10, followed by isocyanurate and phenolic foam insulations with, 8.3 and 7, respectively. They are followed closely by polyurethane and polystyrene insulation at roughly R–6 and R–5. Loose cellulose, fiberglass both blown and in batts, and rock wool both blown and in batts all possess an R-value of roughly 3. Straw bales perform at about R–1.45. Snow is roughly R–1. • Absolutely still air has an R-value of about 5 but this has little practical use: Spaces of one centimeter or greater will allow air to circulate, convecting heat and greatly reducing the insulating value to roughly R–1

9. UCSD: Physics 121; 2011 Convective Heat Exchange • Air (or any fluid) can pull away heat by physically transporting it • really conduction into fluid accompanied by motion of fluid • full, rigorous, treatment beyond scope of this class • General behavior: power convected = P = h·T·A • A is area, T is temperature difference between surface and bath • h is the convection coefficient, units: W/K/m2 • still air has h  2–5 W/K/m2 • higher when T is higher: self-driven convective cells • note that h = 5.67 is equivalent to R = 1 • gentle breeze may have h  5 –10 W/K/m2 • forced air may be several times larger (h  10–50)

10. UCSD: Physics 121; 2011 Convection Examples • Standing unclothed in a 20 C light breeze • h 5 W/K/m2 • T = 17 C • A  1 m2 • P  (5)(17)(1) = 85 W • Our hut from before: • h 5 W/K/m2 • T = 30 C (if the skin is at the hot temperature) • A  12 m2 • P  (5)(30)(12) = 1800 W

11. UCSD: Physics 121; 2011 Radiative Heat Exchange • The Stephan-Boltzmann law tells us: • P = A(Th4 Tc4) • The Stephan-Boltzmann constant,  = 5.6710-8 W/m2/K4 • in thermal equilibrium (Th = Tc), there is radiative balance, and P = 0 • the emissivity ranges from 0 (shiny) to 1 (black) • “black” in the thermal infrared band (  10 m) might not be intuitive • your skin is nearly black (  0.8) • plastics/organic stuff is nearly black (  0.8–1.0) • even white paint is black in the thermal infrared • metals are almost the only exception • for small T around T, P  4AT3 T = (4T3)·A·T • which looks like convection, with h = 4T3 • for room temperature, h  5.7 W/K/m2, so similar in magnitude to convection

12. UCSD: Physics 121; 2011 Radiative Examples • Standing unclothed in room with 273 C walls • and assume emissivity is 0.8 for skin • A 1 m2 • T = 310 K • P  (0.8)(1)(5.6710-8)(3104) = 419 W (burr) • Now bring walls to 20 C • T = 17 C • P  (0.8)(1)(5.6710-8)(3104  2934) = 84 W • pretty similar to convection example • note that we brought our cold surface to 94.5% the absolute temperature of the warm surface, and only reduced the radiation by a factor of 5 (not a factor of 18): the fourth power makes this highly nonlinear

13. UCSD: Physics 121; 2011 Combined Problems • Two-layer insulation • must compute temperature at interface • Conduction plus Convection • skin temperature must be solved • Conduction plus Radiation • skin temperature must be solved • The whole enchilada • conduction, convection, radiation

14. UCSD: Physics 121; 2011 Two-Layer insulation • Let’s take our ice-fishing hut and add insulation instead of replacing the wood with insulation • each still has thickness 0.025 m; and surface area 12 m2 • Now have three temperatures: Tin = 20, Tmid, Tout = 10 • Flow through first is: P1 = 1·(Tin  Tmid)·A1/t1 • Flow through second is: P2 = 2·(Tmid  Tout)·A2/t2 • In thermal equilibrium, must have P1 = P2 • else energy is building up or coming from nowhere • We know everything but Tmid, which we easily solve for: Tmid(1A1/t1 + 2A2/t2) = 2A2Tin/t2 + 2A2Tout/t2 • find Tmid = 9.412 or Tmid = 19.412 depending on which is interior or exterior • heat flow is 282 W (compare to 288 W before: wood hardly matters)

15. UCSD: Physics 121; 2011 Conduction plus Convection • Let’s take our hut with just wood, but considering convection • The skin won’t necessarily be at Tout • Again, thermal equilibrium demands that power conducted through wall equals power wafted away in air • P = h·(Tskin  Tout)·A = ·(Tin  Tskin)·A/t • for which we find Tskin = (Tin/t + hTout)/(h + /t) = 16.7 C • so the skin is hot • P = (5)(26.7)(12)  1600 W • So a space heater actually could handle this (no radiation) • lesson: air could not carry heat away fast enough, so skin warms up until it can carry enough heat away—at the same time reducing T across wood • h may tend higher due to self-induced airflow with large T • also, a breeze/wind would help cool it off

16. UCSD: Physics 121; 2011 Convection plus Radiation • How warm should a room be to stand comfortably with no clothes? • assume you can put out P = 100 W metabolic power • 2000 kcal/day = 8,368,000 J in 86400 sec  100 W • P = h·(Tskin  Tout)·A + A(Tskin4 Tout4)  (hA + 4AT3)T • with emissivity = 0.8, T = 293 K • 100 = ((5)(1) + 4.56)T • T = 10.5 • so the room is about 310  10.5 = 299.5 K = 26.5 C = 80 F • iterating (using T = 299.5); 4.56  4.87; T  10.1 • assumes skin is full internal body temperature • some conduction in skin reduces skin temperature • so could tolerate slightly cooler

17. UCSD: Physics 121; 2011 The whole enchilada • Let’s take a cubic box with a heat source inside and consider all heat transfers • P = 1 W internal source • inside length = 10 cm • thickness = 2.5 cm • R-value = 5 • so 5.67t/ = 5   = 0.028 W/m/K • effective conductive area is 12.5 cm cube  Ac = 0.09375 m2 • external (radiative, convective) area is 15 cm cube  Aext = 0.135 m2 • assume h = 5 W/K/m2,  = 0.8, Text = 293 K • assume the air inside is thoroughly mixed (perhaps 1 W source is a fan!)

18. UCSD: Physics 121; 2011 The enchilada calculation • power generated = power conducted = power convected plus power radiated away P = ·(Tin  Tskin)·Ac/t = (hAext + 4AextT3)·(Tskin  Text) • first get Tskin from convective/radiative piece • Tskin = Text + P/ (hAext + 4AextT3) = 20 + 1.0/(0.675+0.617) • Tskin = 20.8 (barely above ambient) • now the T across the insulation is P·t/(Ac·) = 9.5 • so Tin = 30.3 • Notice a few things: • radiation and convection nearly equal influence (0.617 vs. 0.675) • shutting off either would result in small (but measurable) change

19. UCSD: Physics 121; 2011 Timescales • So far we’ve looked at steady-state equilibrium situations • How long will it take to “charge-up” the system? • Timescale given by heat capacity times temperature change divided by power •   cp·m·T/P • For ballpark, can use cp 1000 J/kg/K for just about anything • so the box from before would be 2.34 kg if it had the density of water; let’s say 0.5 kg in truth • average charge is half the total T, so about 5 • total energy is (1000)(0.5)(5) = 2500 J • at 1W, this has a 40 minute timescale

20. UCSD: Physics 121; 2011 Heating a lump by conduction • Heating food from the outside, one relies entirely on thermal conduction/diffusion to carry heat in • Relevant parameters are: • thermal conductivity,  (how fast does heat move) (W/m/K) • heat capacity, cp (how much heat does it hold) (J/kg/K) • mass, m (how much stuff is there) (kg) • size, R—like a radius (how far does heat have to travel) (m) • Just working off units, derive a timescale: •   (cp/)(m/R)  4(cp/)R2 • where  is density, in kg/m3:   m/((4/3)R3)  m/4R3 • faster if: cp is small,  is large, R is small (these make sense) • for typical food values,   6 minutes  (R/1 cm)2 • egg takes ten minutes, turkey takes 5 hours

21. UCSD: Physics 121; 2011 Lab Experiment • We’ll build boxes with a heat load inside to test the ideas here • In principle, we can: • measure the thermal conductivity of the insulation • see the impact of emissivity changes • see the impact of enhanced convection • look for thermal gradients in the absence of circulation • look at the impact of geometry on thermal state • see how serious heat leaks can be • Nominal box: • 10 cm side, 1-inch thick, about 1.5 W (with fan)

22. UCSD: Physics 121; 2011 Lab Experiment, cont. • We’ll use power resistors rated at 5 W to generate the heat • 25  nominal • P = V2/R • At 5 V, nominal value is 1 W • can go up to 11 V with these resistors to get 5 W • a 12  version puts us a bit over 2 W at 5 V • Fans to circulate • small fans operating at 5 V (and about 0.5 W) will keep the air moving • Aluminum foil tape for radiation control • several varieties available • Standard building insulation

23. UCSD: Physics 121; 2011 Lab Experimental Suite

24. UCSD: Physics 121; 2011 Temperature measurement • There are a variety of ways to measure temperature • thermistor • RTD (Resistive Temperature Device) • AD-590 • thermocouple • Both the thermistor and RTD are resistive devices • thermistor not calibrated, nonlinear, cheap, sensitive • platinum RTDs accurate, calibrated, expensive • We’ll use platinum RTDs for this purpose • small: very short time constant • accurate; no need to calibrate • measure with simple ohm-meter • R = 1000.0 + 3.85(T  0C) • so 20C would read 1077.0 

25. UCSD: Physics 121; 2011 Random Notes • Rig fan and resistor in parallel, running off 5V • fan can accept range: 4.5–5.5 V • if you want independent control, don’t rig together • Use power supply current reading (plus voltage) to ascertain power (P = IV) being delivered into box • Make sure all RTDs read same thing on block of thermally stabilized chunk of metal • account for any offset in analysis • Don’t let foil extend to outside as a cold finger • Make sure you have no air gaps: tape inside and out of seams • but need to leave top accessible • nice to tape fan to top (avoid heat buildup here) • can hang resistor, RTD from top as well (easy to assemble)

26. UCSD: Physics 121; 2011 Random Notes, continued • Measure temp. every ~2 minutes initially • tie white leads of RTDs to common DVM all together • label red lead so you know where it goes • After equilibrium is reached, measure skin temperatures • hold in place with spare foam (not finger or thermal conductor!) • best to note time of each digit change • allows extrapolation to final (slow otherwise) • We have limited RTDs, so 3−4 per group will be standard • locate inside RTD in fan exhaust, so representative • use external RTD for ambient, skin (double duty) • some experiments will want more RTDs (gradients) • Once equilibrated, go to configuration B • turn on external fan, coat with foil, poke a hole, cold finger

27. UCSD: Physics 121; 2011 Random Notes, continued • Send your data points to me via e-mail so I can present the amalgam of results to the class • use format: hh:mm RTD1 RTD2 RTD3 etc. • example: 11:43 1088 1155 1152 1228 • include a description of what each column represents • Also include basic setup and changes in e-mail so I know what I’m plotting • Also include in the message temperatures you measure only once, or occasionally (like skin temp.) • I’ll make the data available for all to access for the write-ups

28. UCSD: Physics 121; 2011 Example Series

29. UCSD: Physics 121; 2011 Temperature differences

30. UCSD: Physics 121; 2011 References and Assignment • Useful text: • Introduction to Heat Transfer: Incropera & DeWitt • Reading in text: • Chapter 8 (7 in 3rd ed.) reading assignment: check web page for details

31. UCSD: Physics 121; 2011 Thermal Building Design • You can get R-values for common construction materials online • see http://www.coloradoenergy.org/procorner/stuff/r-values.htm • Recall that R = 5.67t/ • so power, P = 5.67AT/R • Composite structures (like a wall) get a net R-value • some parts have insulation, some parts just studs • if we have two areas, A1 with R1 and A2 with R2, total power is P = 5.67A1T/R1 + 5.67A2T/R2 • so we can define net R so that it applies to Atot = A1 + A2 • 1/Rtot = (A1/Atot)/R1 + (A2/Atot)/R2 • in example on web site, studs take up 15%, rest of wall 85% • P = 5.67AtotT/Rtot

32. UCSD: Physics 121; 2011 Handling External Flow as R-value • On the materials site, they assign R-values to the air “layer” up against the walls • outside skin R = 0.17 • inside skin R = 0.68 • This accounts for both convection and radiation. How? • recall that power through the walls has to equal the power convected and radiated P = 5.67A(TinTskin)/R = hconvA(TskinTout) + hradA(TskinTout) P = 5.67A(TinTskin)/R = heffA(TskinTout) • where hrad 4T3, and heff = hconv + hrad • We can solve this for Tskin, to find Tskin = (5.67Tin/R + heffTout)/(5.67/R +heff)

33. UCSD: Physics 121; 2011 Putting Together • Inserting the expression for Tskin into the conduction piece, we get: P = 5.67A(TinTskin)/R = 5.67A(Tin(5.67Tin/R+heffTout)/(5.67/R+heff))/R • multiply the solitary Tin by (5.67/R+heff)/(5.67/R+heff) • 5.67Tin/R term cancels out P = 5.67A((heffTinheffTout)/(5.67/R+heff))/R P = 5.67A(TinTout)heff/(5.67+heffR) • which now looks like a standard conduction relation between inside and outside temperatures, with an effective R: Reff = R + 5.67/heff • The effective R is the R-value of the original wall plus a piece from the air that looks like 5.67/heff • the site has interior air layer Reff=0.68, or heff = 8.3, which is appropriate for radiation plus convection • for exterior, they use Reff = 0.17, or heff = 33, representing windy conditions

34. UCSD: Physics 121; 2011 A model house • Ignoring the floor, let’s compute the heat load to keep a house some T relative to outside • useful to formulate G = P/T in W/K as property of house • Assume approx 4040 ft floorplan (1600 ft2) • 8 feet tall, 20% windows on wall • Wall: 100 m2, windows: 20 m2, ceiling: 150 m2, roof 180 m2 • Can assess for insulation or not, different window choices, etc. • Gwindow = 125, 57, 29 for single, double, or deluxe window • Gwall = 142, 47 for no insul, insul • Gceil = 428, 78 for no insul, insul • Groof = 428, 90 for no insul, insul

35. UCSD: Physics 121; 2011 Dealing with the Ceiling • The Gceil and Groof require interpretation, since the T across these interfaces is not the full T between inside and outside • there is a Tattic in between • but we know that the heat flow through the ceiling must equal the heat flow through the roof, in equilibrium • so Gceil(TinTattic) = Groof(TatticTout) • then Tattic = (GceilTin+GroofTout)/(Gceil+Groof) • so that Gceil(TinTattic) = Gup(TinTout) • where Gup = GceilGroof/(Gceil+Groof), in effect acting like a parallel combination • So Gup evaluates to: • Gup = 214, 74, 66, 42 for no/no, ceil/no, no/roof, ceil/roof insulation combinations

36. UCSD: Physics 121; 2011 All Together Now • The total power required to stabilize the house is then Ptot = Gtot T, where Gtot = Gwindow + Gwall + Gup • For a completely uninsulated house: • Gtot = 481 W/K • requires 7.2 kW to maintain T = 15C • over 5 months (153 days), this is 26493 kWh, costing $2649 at $0.10/kWh • Completely insulated (walls, ceiling, roof, best windows), get Gtot = 118 W/K • four times better! • save $2000 per cold season (and also save in warm season)